Existence and Uniqueness of Positive Solutions for a Class of Nonlinear Fractional Differential Equations with Singular Boundary Value Conditions

)is paper focuses on a singular boundary value (SBV) problem of nonlinear fractional differential (NFD) equation defined as follows: D0+υ(τ) + f(τ, υ(τ)) � 0, τ ∈ (0, 1), υ(0) � υ′(0) � υ′′(0) � υ′′(1) � 0, where 3< β≤ 4, D β 0+ is the standard Riemann–Liouville fractional (RLF) derivative. )e nonlinear function f(τ, υ(τ)) might be singular on the spatial and temporal variables. )is paper proves that a positive solution to the SBV problem exists and is unique, taking advantage of Green’s function through a fixed-point (FP) theory on cones and mixed monotone operators.

Unlike the existing studies, two novel features are induced in this paper. (1) To the best of our knowledge, boundary condition (5) is firstly considered for NFD. (2) Conditions (H2) and (H5) imposed on f are different from those in [26][27][28][29][30][31][32][33][34]. e remainder of this paper is structured as follows. Preliminaries are given in Section 2, including definitions, lemmas, the deduction of Green's function for problem (4), and new positive properties. Section 3 proves the presence of positive solutions of (4) by the Guo-Krasnoselskii FP theory and demonstrates an example. Section 4 discusses the uniqueness of the positive solution of (4) by a mixed monotone operator and demonstrates another example.

Preliminaries
e lemmas and definitions from [3] are given for the convenience of the reader as follows: Definition 1 (see [3]). e RLF integral of the order β > 0 of a function f(x): (0, +∞) ⟶ R is formulated: provided the right side is pointwise defined on (0, +∞).
e solution of the NFD equation defined as where N is the smallest integer greater than or equal to β and μ ∈ C(0, 1) ∩ L(0, 1).
e solution of (10) is unique and as follows: where Green's function G(τ, s) is denoted as Proof. (10) is rewritten as follows through Lemma 2: where C i ∈ R, i � 1, 2, 3, 4. From the boundary conditions Mathematical Problems in Engineering By the condition u ″ (1) � 0, we have Accordingly, the unique solution of problem (10) and (11) is given as Lemma 3 is proved with this. □ Lemma 4. e properties of G(τ, s) defined by (13) are as follows: Proof. Property (4) is obvious and (3) holds from (1). us, here (1) and (2) will be proved.

Mathematical Problems in Engineering
From (18)-(21), we have the first conclusion in Lemma 4 which holds.

Lemma 6.
e unique fixed point x * of A exists when a constant c ∈ (0, 1) satisfies where A is a mixed monotone operator.

Presence of Positive Solutions of SVB
e presence and multiplicity of positive solutions of (4) and (5) is investigated here.
For a Banach space Ψ � C[0, 1] with the maximum norm max 0≤τ≤1 |μ(τ)|, let K ∈ Ψ denote a nonnegative cone defined as e operator T is defined as follows: e following are assumed for later use: and for any r > 0, Lemma 7. For any r > 0, T: K∖B r ⟶ K is completely continuous.
From (H 2 ) and (1) of Lemma 4 meaning that T is well defined. And, by (1) of Lemma 4, So, T maps K, B r into K. For a bounded set D ∈ K, B r , a number R exists such that R > r and r ≤ ‖μ‖ ≤ R for any μ ∈ D. So, we have which means T(D) is uniformly bounded. G(τ, s) is uniformly continuous on [0, 1] × [0, 1]. Accordingly, δ > 0 exists for any ε > 0, such that|τ Consequently, implying T(D) is equicontinuous. According to the Arzela-Ascoli theorem, T: K∖B r ⟶ K is compact.